Journal of Engineering Physics and Thermophysics, Vol. 86, No. 6, November, 2013
HEAT AND MASS TRANSFER IN DISPERSED AND POROUS MEDIA IMPROVING THE PERFORMANCE OF AN ADSORPTION HEAT CONVERTER IN CONDENSATION AND EVAPORATION OF THE ADSORBATE IN SORBENT PORES M. Yu. Lyakh, O. S. Rabinovich, L. L. Vasiliev, and A. P. Tsitovich
UDC 621.576
The possibilities of raising the specific refrigerating capacity and the cooling temperature of an adsorption refrigerator through the phase transition of the adsorbate in low-temperature-sorbent pores have been investigated by the computer-modeling method. Using an adsorption refrigerator with busofite-based MnCl2 and BaCl2 sorbents (in the high-temperature and low-temperature adsorbers respectively) as an example, it has been shown that the operating regime of the refrigerator with adsorbate condensation and evaporation enables one to raise the specific capacity of the apparatus by 20% and to double the average cooling temperature. Keywords: adsorption heat converter, adsorption, capillary condensation, composite sorbent. Introduction. Investigation of sorption converters of thermal energy with the aim of improving their performance is the focus of quite a number of scientific works. Some publications propose schemes of improved adsorption cycles which include additional stages apart from standard operations. For example, the use of mass recovery in addition to a standard adsorption cycle causes the coefficient of performance to grow [1, 2]. Increase in the number of adsorbers enables one to implement continuous processes of cooling and heating and hence to increase the obtained heat output [3]. The use of the thermal-wave effect furnishes analogous results [4]. The performance of adsorption and thermochemical converters of thermal energy can also be improved by utilizing novel modified composite sorbents [5, 6]. However, certain possibilities of upgrading sorption heat converters remain to be studied. In particular, of interest is the idea of the phase transition effected directly inside the adsorption unit, which the authors came up with in [7]. The process of condensation and evaporation enables one to utilize the heat of phase transition to produce an additional temperature jump in the adsorber, which leads to a considerable increase or reduction in the outlet temperature of the heat-transfer agent. The idea in question corresponds in a sense to combining an adsorber, a condenser, and an evaporator in one module, in contrast to traditional diagrams of adsorption heat converters with separated units of the indicated types. Apart from the increase in the temperature effect of heat conversion, the proposed technique must enable one to raise the specific power of the heat converter and to reduce its specific consumption of materials and dimensions. The present work seeks to theoretically analyze the indicated idea and the potential advantages provided by its realization. Formulation of the Problem. Consideration is given to the influence of the condensation and evaporation of an adsorbate in the pores of a sorbent of a low-temperature adsorber on the performance of an adsorption refrigerator whose diagram is presented in Fig. 1. The basic elements of the refrigerator are low-temperature and high-temperature adsorbers connected by a gas tube on which there is a throttle making it possible to control the sorbate flow between the reactors. A heat-transfer agent from two heat reservoirs with high Thigh and low Tlow (of the order of room) temperatures can be fed to the adsorbers; here, the two heat reservoirs are connected to the high-temperature adsorber, and one reservoir with temperature Tlow is connected to the low-temperature adsorber. With account taken of the sorbate condensation in the low-temperature adsorber, the capacities of the reactors must be matched. Both adsorbers are assumed to be identical in structure and to have the shape of cylinders of the same diameter. Therefore, the adsorbers’ capacities are matched due to the change in the length of the high-temperature adsorber as a function of the amount of the condensate in the low-temperature one, and the ratio of the lengths of the high-temperature and the low-temperature adsorbers is used for analysis of results. A diagram of an individual A. V. Luikov Heat and Mass Transfer Institute, National Academy of Science, 15 P. Brovka Str., Minsk, 220072, Belarus; email:
[email protected]. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 86, No. 6, pp. 1185–1198, November–December, 2013. Original article submitted July 26, 2013. 0062-0125/13/8606-1259 ©2013 Springer Science+Business Media New York
1259
Fig. 1. Diagram of the adsorption refrigerator under study.
adsorber (reactor) is depicted in Fig. 2. A liquid heat-transfer agent is fed by the central channel of the reactor. A liquid having the specific heat of water but possessing a lower freezing point, e.g., water with addition of salt or alcohol, is considered as the heat-transfer agent. The adsorbate is fed from the outside of the adsorber. Both adsorbers are assumed to be filled with composite sorbent consisting of carbon fiber (busofite), which ensures physical adsorption and in whose pores the adsorbate may condense, and of salt on which chemical adsorption occurs. Use is made of MnCl2 as the high-temperature salt and of BaCl2 as the low-temperature salt. The working medium (sorbate) is ammonia. The problem on the dynamics of distributions of the adsorbate temperature and concentration in each reactor (adsorber) is solved in a one-dimensional formulation under the assumption that the sought quantities depend solely on the radial coordinate. This simplified formulation of the problem assumes that the gas moves only in the radial direction and there are no end effects. In formulating a mathematical model of a heat converter, we made the following assumptions: 1) diffusion resistance of the adsorbent is low; 2) the temperatures of solid, liquid, and gaseous phases are equal at each point because of the high value of the coefficient of volume interphase heat transfer; 3) free gas in the adsorbers is ideal; 4) there is no expenditure of thermal energy for compressing or expanding the gas; 5) gravity forces are negligible, since capillary pressure produced in the largest macropores is much higher than the hydrostatic pressure in the adsorber (at rp = 100 μm, Pσ ~ σ/rp ≈ 500 Pa and Ph = ρgh ≈ 100 Pa); 6) filtration limitations are disregarded, since the capillary pressure in mesopores is at least one order of magnitude higher than the pressure produced in the reactor, and filtration is minor in macropores (in the case of their partial filling we also do not have significant pressure differences over the adsorption bed); therefore, the pressure distribution in the adsorber is considered uniform. Mathematical Model. In accordance with the assumptions made, we have created a mathematical model of an adsorption refrigerator enabling us to take account of the capillary condensation of the adsorbate. The model’s components responsible for adsorption, thermal, and mass-transfer processes have been described in [8, 9]; however, for generality of presentation we give a description of the model in complete form. As has been mentioned above, a sorbent represents a composite material whose components exhibit chemical and physical adsorption. To calculate chemical adsorption we use the following kinetic equations [10] for the decomposition reaction: dx = Ax C dt
and for synthesis reaction:
1260
Peq ⎞ ⎛ ⎜1 − ⎟, P ⎠ ⎝
Fig. 2. Diagram of a single adsorber: 1) zone of adsorbate feed; 2) sorbent; 3) heat-transfer agent.
dx C = A (1 − x) dt
Peq ⎞ ⎛ ⎜1 − ⎟. P ⎠ ⎝
The value and rate of this kind of adsorption is determined as ach = amax x,
Wch = amax
dx . dt
The pressure corresponding to the equilibrium state of each sorbent is determined from the relation Rg ln Peq = −
ΔH + ΔS . T
(1)
To find the equilibrium value of physical adsorption we use the Dubinin–Radushkevich equation [11]
aeq
⎛ W0 T2 ⎜ = exp ⎜ − B 2 b β ⎜ ⎝
⎡ ⎢ln ⎢ ⎣⎢
2 ⎛ ⎛ T ⎞ 2 P ⎞ ⎤ ⎞⎟ cr ⎜ ⎟ ⎥ , ⎜ ⎝⎜ Tcr ⎠⎟ Pg ⎟ ⎥ ⎟⎟ ⎝ ⎠ ⎦⎥ ⎠
(2)
where b is the constant from the Van der Waals equation. The physical adsorption rate is determined from the kinetic equation [12] daph dt
= Wph = K s0 exp (− E RgT ) ( aeq − aph ) ,
(3)
where Ks0 is the preexponential factor. In the case of the joint course of two types of adsorption in the sorbent the total amount of adsorbed substance per unit of the utilized composite sorbent is computed by the additive method:
a = aph f ph + ach f ch ,
(4)
where f is the mass content of the chemical or physical sorbent in the composite sorbent. The total adsorption rate is determined analogously. As has already been noted, adsorbate condensation may appear in adsorbent pores in the process of adsorption under certain conditions. No condensation occurs in micropores (deff < 2nm), since adsorption in such pores is not accompanied by the formation of adsorption beds, and we have the volume filling of the adsorption space with gas. In mesopores (2 nm < deff < 50 nm), first we have mono-and polymolecular adsorption on the surface and thereafter volume filling by the mechanism of capillary condensation. No capillary condensation is implemented in macropores (deff > 50 nm), since the size of such pores is too large and the corresponding capillary pressure is low. The gas condenses in such pores only 1261
on attainment of thermodynamic parameters corresponding to the region of condensed state. Thus, capillary condensation may occur only in mesopores. This phenomenon appears on condition of wettability of the pore walls with the adsorbed phase, when the pressure of the adsorbate’s saturated steam above the curved interface in a pore is lower than the gas pressure and the temperature is lower than the critical temperature [13]. It follows that capillary condensation in mesopores occurs even before the thermodynamic parameters find themselves in the region of condensed state. As the pressure of the sorbed gas grows, condensation in the porous body is continuous, finally filling first small pores (mesopores) and thereafter larger pores (macropores) with liquid phase; we have capillary sorbate condensation in mesopores and a regular phase transition in macropores. In the present work, we use a simplified assumption of the binary pore distribution in the sorbent: there are mesopores with a certain average diameter d mes and a volume fraction εmes and macropores with a volume fraction εmac. Thus, the content of the liquid phase in meso- and macropores is calculated separately, whereas the total content of the liquid in the sorbent is determined as the sum of the indicated quantities. In the calculations given below, we have selected the following structural characteristics of the sorbent which correspond to busofite: d mes = 10 nm and the volume fraction of mesopores εmes = 0.065; the volume fraction of macropores in the sorbent εmac is taken to be 0.35. The intensity of capillary condensation and evaporation is determined by the Hertz–Knudsen equation [14] mes ≈ J gas-liq
( Psat′ (T ) − Pg ) S 2π
RgT
,
gas-liq
(5)
M
The saturated-steam pressure above the concave surface of the condensate is determined from the Kelvin equation ⎛ PV ⎞ ′ = Psat (T ) exp ⎜ − σ l ⎟ , Psat ⎜ RgT ⎟ ⎝ ⎠
(6)
where Pσ = 2σ rp . Condensation in macropores begins when the pressure in the adsorber becomes higher than the saturatedsteam pressure above the plane surface of the liquid. The intensity of this process is defined as mac ≈ J gas-liq
( Psat (T ) − Pg ) S 2π
RgT
gas-liq
.
(7)
M
With allowance for the assumptions made, we write mass equations for the liquid phase and the gas as follows: ε mes
mes ∂θlig ρ lig
∂t
+ ε mac
mac ∂θlig ρ lig
∂t
mes mac = − J gas-liq − J gas-liq ,
(8)
⎛ ∂θgmes ρ g ∂θgmac ρ g ∂ρ g ⎞ mes mac ⎜ ε mes ⎟ + ∇ (ρ g v g ) = J gas-liq + ε mac + (1 − ε s − ε mac − ε mes ) + J gas-liq − ε sρ sWads . ⎜ ⎟ ∂ t ∂ t ∂ t ⎝ ⎠
(9)
In the process of operation of the converter, pressures in the reactors change and their values are determined with allowance for the fact that the integral mass balance dm = w − K thr ( P − P* ) dt
(10)
is satisfied. Here, Kthr is the transmittance of the throttle, m is the ammonia mass in an unadsorbed state, i.e., in the liquid and gaseous phases: m =
(
mac + ε mac ρ g θ gmac + ρ lig θ lig
∫ (ε mes
+ ε mac ) ρ g dV for the high-temperature reactor and m =
V
)) dV
for the low-temperature reactor, and w =
is determined from the equation of balance of thermal energy: 1262
mes ∫ (ε mes (ρg θg
mes + ρ lig θ lig
)
V
∫ ρsWads dV .
V
The temperature in the adsorber
(
)
⎡ ε mes θgmes + ε mac θ gmac + 1 − ε s − ε mes − ε mac ρ g cpg ⎢ ⎣ ∂T mac mes + ε mac θlig + ε mes θ lig ρ lig c Plig + ε s ρ s c Ps + ε s ρ s ac Pa ⎤ ⎦ ∂t
(
)
(
(11)
)
mac mes Qgas-liq + ε s ρ s QsWads . + vg ρ g c Pg ∇T = ∇ (λ eff ∇T ) − J gas-liq + J gas-liq
Initial and Boundary Conditions. It is assumed that, at the initial instant of time, the pressure in all the reactors is the same and equal to atmospheric pressure under normal conditions: Plow
t =0
= Phigh
= P0 .
t =0
The temperature is also the same in all the reactors and is equal to 30oC: Tlow
t =0
= Thigh
t =0
= Tenv .
On the external and internal boundaries of the adsorbers, we specify the following boundary conditions: −λ
∂T ∂r
= 0,
−λ
r=R
∂T ∂r
r = R0
= α (T − Tf,in ) .
Heat-transfer conditions for the heat-transfer agent and the reactor wall are of the form
Gf c Pf (Tf,out − Tf,in ) = αS (Tw − Tf,in ) , hence we obtain an expression for the temperature of the heat-transfer agent at the reactor outlet: Tf,out = Tf,in +
αS (Tw − Tf,in ) . Gf c Pf
(12)
In subsequent calculations, we select the greatest possible coefficient of heat exchange between the heat-transfer agent and the adsorber, i.e., a coefficient such that the outlet temperature of the heat-transfer agent is equal to the temperature of the interior adsorber wall. Calculation of Efficiency Characteristics. The principle indices used for determining heat-conversion efficiency is the coefficient of performance (COP) and the specific cooling power (SCP). In the present work, these characteristics are defined as follows [10]:
COP =
( (T
) ) dt⎦⎤
low low ⎡ α low S low Tf,in − Tf,out dt ⎤ ⎣∫ ⎦ Plow
⎡ α high S high ⎣∫
high f,in
high − Tf,out
Phigh
,
SCP =
(
)
low low ⎡ α low S low Tf,in − Tf,out dt ⎤ ⎣∫ ⎦ Plow
Δtcyc mslow
,
where Plow and Phigh refer to the stages of low and high pressures. We emphasize that the specific refrigerating capacity of a heat converter is calculated from the time of a complete cycle including both stages of the process: the high-pressure stage and the low-pressure stage. This method of SCP determination yields lower values than calculation from the time of the low-pressure stage, which should be taken into account when the obtained results are compared to the works of some authors using an alternative calculation method. In determining the efficiency of thermal-energy conversion, evaluation of the temperature of the heat-transfer agent at exit from the low-temperature reactor is important. In the work, this characteristic is calculated from the difference of the temperatures of the low-temperature heat-transfer agent at this reactor’s inlet and outlet, which is averaged over the period of
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Fig. 3. Time dependences of the temperature on the surface of the low-temperature reactor (1), of the temperature inside the liquid line of cooling of the low-temperature sorber (2), and of the difference of the temperatures ΔT of the heat-transfer agent at the reactor inlet and outlet: solid curves, calculation; dashed curves, experiment. cooling (low-pressure stage). The characteristic in question is called, for simplicity, the average cooling temperature and is determined from the following formula:
(
)
low low ⎡ Tf,in − Tf,out dt ⎤ ⎣∫ ⎦ Plow ΔT = . Δt Plow
(13)
The indicated characteristics of efficiency are mainly determined by the initial and final temperatures of the heattransfer agents. Verification of the Model. The above-presented model was tested by comparing calculation results and experimental data given in [7]. The diagram of the adsorption refrigerator used in the experiments in [7] is analogous to the diagram in Fig. 1. The only difference is that in the experiment we used two adsorbers identical to the low-temperature adsorber in dimension instead of one long high-temperature adsorber. Figure 3 gives the dynamics of the average temperatures of the adsorbers and the difference of the temperatures of the heat-transfer agent at the inlet and outlet of the low-temperature reactor, obtained experimentally in [7] and numerically from the presented model. An analysis of the results in Fig. 3 points to minor disagreements between experimental and calculated data, with the theory and experiment yielding the same minimum surface temperatures of the low-temperature reactor: –14oC. It should be noted that experiments with an adsorption refrigerator [7] were conducted in the region of parameters at which capillary condensation is realized to a small extent. Thus, for the considered case, the calculated fraction of the condensate in mesopores amounts to 6%, and in macropores, to only 0.5%. For this reason, verification of the model for regimes with a large filling of the pores with condensate remains the objective of subsequent experimental investigations. Also, we have calculated the SCP for the considered case. In the experiment, this quantity was 27 W/kg in an operating cycle of the setup; numerical modeling yields quite a similar result: 36.8 W/kg. More significant differences between experimental and calculated data have been revealed for the specific refrigerating capacity of the heat converter in a time interval corresponding to the production of one-eighth of the maximum quantity of refrigeration: 107 and 180 W/kg respectively. This difference in the rates of the initial step of cooling may be due to both the inaccurate description of the kinetics of adsorption and phase transition and the non-one-dimensionality of actual processes in the heat converter. Modeling Results. Basic Controlling Parameters and Dynamics of the Evaporation and Condensation. The operating cycle of the two-adsorber refrigerator in question consists of two basic stages: low-pressure and high-pressure stages. Figure 4 gives diagrammatically the operating principle of such a refrigerator. In the first step, i.e., the high-pressure stage, a high-temperature heat-transfer agent is supplied to the high-temperature reactor, whereas the low-temperature adsorber is cooled from the low-temperature reservoir. Because of this, the working gas (adsorbate) is desorbed in the high-temperature reactor initially totally saturated with adsorbate and enters the neighboring reactor where it is adsorbed. In the second step, the process of adsorption is implemented in the reactor with a high-temperature sorbent and the process of desorption is implemented in the low-temperature unit. During the low-pressure stage, a low-temperature heat-transfer agent is supplied
1264
Fig. 4. Diagram of operation of the adsorption refrigerator at the stages of high (a) and low (b) pressures.
to the two absorbers. It is precisely in this step that the required cooling effect is attained as a result of the reduction in the temperature in the low-temperature reactor due to the desorption. As has already been noted, on the tube on which the adsorbate is transferred from one adsorber to the other, a throttle is installed using which one can control the gas flow. In the calculations, the transmittance of the throttle Kthr is selected so that the created rate of adsorbate transfer between the reactors corresponds to the greatest possible rate of consumption or release of the gas in the adsorbers. If the transmittance set is too small, the process of heat conversion will be retarded, which will cause the SCP to decrease. When the Kthr value is far too high, there can appear numerical instability due to the upset of mass balance between the adsorbers. A constant fairly high transmittance of the throttle was used for the entire cycle of heat conversion in the calculations. Test calculations confirmed the assumption that the decrease in Kthr at any stage of the cycle (e.g., cutoff by the throttle in the period of heating of the adsorber) only leads to a reduction in the SCP and in the temperature of the heat-transfer agent at exit from the low-temperature reactor. The operation of the adsorption thermal-energy converter was modeled at the following parameters: Gf = 0.003 kg/s, Kthr = 1·10–9 m·s, Llow = 1 m, R = 0.0245 m, and R0 = 0.008 m. Sorbents Busofite + MnCl2 and Busofite + BaCl2 with parameters cРs = 1154 and 924 J/(kg⋅K) and ρs = 285 and 362 kg/m3 respectively were utilized. The heat capacity of the adsorbate cРa on the indicated sorbents was taken to be 1657 and 1470 J/(kg⋅K). The kinetics of chemical adsorption on MnCl2 and BaCl2 salts was determined by the parameters Asyn = 0.001019 and 0.0125 s–1, Adec = 0.0028 and 0.0195 s–1, Сsyn = 1.185 and 2.104, Сdec = 1 and 1.005, ΔH = 47.4 and 37.7 kJ/mole, and ΔS = 228.1 and 227.3 J/(mole⋅K) [15], whereas the kinetics of physical adsorption was determined by the parameters B/β2 = 1.61·10–6 К–2 , E/Rg = 2812, and Ks0 = 0.075 s–1 [16]. The adsorption heat on Busofite was Qs = 3300 kJ/kg. The properties of ammonia were determined by the parameters cРg = 2289, cРliq = 4827 J/(kg·К), ρliq = 595 kg/m3 and σ = 0.23321 N/m. The enthalpy and the entropy of ammonia evaporation were taken to be 23.4 kJ/mole and 192 J/(mole⋅K) respectively. To illustrate the dynamics of condensation and evaporation processes in the low-temperature adsorber, Fig. 5 gives the condensate distributions in meso- and macropores at different instants of time of the operating cycle. These data correspond to the case where the heat-reservoir temperatures are Tlow = 15оС and Thigh = 150оС, and the ratio of the reactor lengths is Lhigh/Llow = 4. Under these conditions, the pores of the sorbent of the low-temperature reactor are completely filled with liquid phase at the high-pressure stage. Red-color curves in Fig. 5 correspond to the capillary condensation, whereas green-color curves, to the condensation in macropores. The time in each figure corresponds to the period that has passed from the beginning of the cycle. Figure 5 a–d depicts in sufficient detail the condensation dynamics. It is seen that mesopores are filled much more rapidly than macropores. This is due to the fact that the share of mesopores is much smaller than that of macropores: 6.5 and 35% respectively. Also, it follows from the plots in this figure that condensation in the adsorber is in the direction from the interior wall to the exterior wall due to the fact that the heat-transfer agent supplied to the reactor cools it on the inside, thus creating conditions for the condensation in this part of the reactor. Comparing the results in Fig. 5a–d and e–h, we can infer that in the operation of the refrigerator under study the process of condensation is much slower than the evaporation. This is attributable to the fact that at the high-pressure stage, when condensation occurs, the pressures in the reactors are equalized to a large extent and turn out to be low for intense condensation. As far as the process of evaporation is concerned, at the instant it begins we have a sharp reduction in the pressure in the reactors when the heat reservoirs feeding the heat-transfer agent to the high-temperature reservoir are switched, 1265
Fig. 5. Condensate distribution in the meso- and macropores of the sorbent on the radius of the low-temperature adsorber at different instants of time at the stages of condensation (a–d) and evaporation (e–h): a) t = 1000, b) 3000, c) 6000, d) 10,000, e) 11,800, f) 12,200, g) 12,600, and h) 13,000 s.
and the rate of the process is high. It follows from Fig. 5e–h that the evaporation in macropores gradually begins in the entire reactor; however, the process is much more intense in its interior part due to the heating of the reactor from the pumped heat-transfer agent. Evaporation in mesopores, as condensation, is in the direction from the interior wall to the exterior wall, since the heat-transfer agent supplied to the reactor at this stage performs the function of heating. Presumably, ammonia in mesopores, evaporated from the reactor’s interior part, enters its exterior part via macropores. Influence of the Parameters of the Adsorption Refrigerator Operating under the Conditions of Adsorbate Condensation in the Low-Temperature Adsorber on Its Performance Characteristics. The adopted model of condensation and evaporation of the adsorbate in the low-temperature unit is based on the assumption of independence of phase transitions and of adsorption and desorption of the adsorbate. This assumption is mainly based on the separation of the indicated processes in different classes of pores, which is true of certain types of sorbents. Thus, for such microporous sorbents as busofite, the bulk of the adsorbed gas (more than 80%) is concentrated in micropores, whereas phase transitions are effected in mesoand macropores. The range of adequacy of the model in the aspect in question is limited by the condition of absence of the blocking of gas transport to the micropores or from them as a result of the sorbate condensation. This condition is ensured 1266
Fig. 6. Characteristics of the efficiency of the adsorption heat converter COP (a, d, and g), SCP (b, f, and h), and 〈ΔTf〉 (c, f, and i) under different conditions of switching of the stages of a heat-conversion cycle: a, b, and c) Thigh = 140 and Lhigh/Llow = 1; d, e, and f) 170 and 1, and g, h, and i) 170oC and 2.55. by the circumstance that, as the modeling shows, condensation at the high-pressure stage begins, once the adsorption in the pores is totally completed, in practice, and conversely, desorption at the low-pressure stage starts once the sorbate evaporation is completed. Furthermore, as is clear from Fig. 5, there is no blocking of transport of the gas evaporated in mesopores at the low-pressure stage, since free macropore space is always left for gas filtration. In connection with fact that the present work seeks to investigate the influence of adsorbate condensation and evaporation in the pores of a low-temperature adsorber on the performance of an adsorption thermal-energy converter, first we consider the methods of control of the phase transition in the adsorber. The onset of the condensation of the adsorbate in the low-temperature adsorber is determined by the following factors (at fixed parameters of sorbents): a) by the temperatures of low-temperature and high-temperature heat reservoirs (with a certain selection of their temperatures, we can create conditions under which condensation does not appear in the low-temperature adsorber or is realized only in mesopores or simultaneously in meso- and macropores); b) by the relation of the adsorption and condensation capacities of the reactors as far as the amount of the adsorbate is concerned (in our case by the relation of the reactor lengths) and by the size of meso- and macropores and their volume fractions; c) by the algorithm of switching between the high- and low-pressure stages (by the conditions of switching of reservoirs from which the heat-transfer agent is fed to the high-temperature adsorber). The calculation results given below largely correspond to the temperature of the high-temperature thermal reservoir Thigh = 170оС which ensures the complete filling of the meso- and macropores of the low-temperature adsorber with condensate (at the ratio of the adsorber lengths Lhigh/Llow = 4). As will be shown below (Fig. 6), this condition cannot be obtained at Thigh = 140оС. Selection of the conditions of switching of cyclic stages is of importance for attaining a high heat-conversion efficiency. In the present work, we use conditions under which switching from one stage to the other depends on the value of the difference of the temperatures of the heat-transfer agent at the inlet and outlet of one reactor or another. When the high-pressure stage is switched to the low-pressure stage, account is taken of the temperatures of the high-temperature heat-transfer agent; on completion of the operating cycle (switching from the low-pressure stage to the high-pressure stage), account is taken of the temperatures of the low-temperature heat-transfer agent at the inlet and outlet of the low-temperature adsorber. This 1267
Fig. 7. Time dependences of the temperatures in the adsorbers averaged over the reactor volume (a) and of the temperatures of the heat-transfer agents at the reactor outlet (b) for the case without condensation. selection has been made with the aim of reducing the duration of the cycle, since at the end of any stage, when temperatures in the reactors become nearly equal to the temperatures of the supplied heat-transfer agents, all processes in the adsorbers are retarded, which causes the SCP and the average cooling temperature to decrease. Selection of the temperature differences is based on a comparison of the characteristics of efficiency at different values of these temperatures. Figure 6 gives the isolines of the effective parameters for three different cases corresponding to different values of the heat-reservoir temperatures and the reactor lengths. As is seen, the COP decreases with increase in the differences of the temperatures of the heat-transfer agents at the inlet and outlet of the reactor, but an increase in the SCP is observed. The difference of the temperatures of the high-temperature heat-transfer agent does not influence, in practice, the average cooling temperature; however, this characteristic grows with the temperature difference of the low-temperature heat-transfer agent. Since the selection between the attainment of SCP or ΔTf maxima depends on a concrete problem, we have selected the conditions of switching of the cyclic stages ΔTf,high = 1оС and ΔTf,low = 3оС for all further calculations. Under these conditions, we can obtain fairly high SPC and average cooling temperature, slightly decreasing the COP. Let us consider in greater detail two different cases of operation of an adsorption refrigerator: 1) the refrigerator operates under conditions where no condensation appears in the sorbent pores; 2) the condensate fills meso- and macropores alike completely. In these cases the process of condensation is controlled by matching the capacities of the high-temperature and low-temperature adsorbers, and the temperatures of the heat-transfer agents (170 and 15оС) are selected so that in the case with condensation the pores are completely filled with condensate. First we consider the case without condensation. Figure 7a gives the time dependences of the adsorbers’ temperature whose values are averaged over the entire volume of the reactor. Figure 7b shows the change in the temperature of the heat-transfer agents at the reactors’ outlets with time. These dependences characterize the basic result of operation of a heat engine and can be used to calculate its performance. As indicated above, the operating cycle of the refrigerator in question consists of the stages of high (region I) and low (region II) pressures. At stage I, a low-temperature heat-transfer agent (15оС) is supplied to the low-temperature adsorber, and the high-temperature reactor is heated from the high-temperature heat reservoir (170оС). The process of desorption occurs in this period in the high-temperature reactor, and the process of adsorption accompanied by the release of heat occurs in the low-temperature reactor. Switching between the stages is effected by changing the heat reservoir from which the heat-transfer agent is fed to the high-temperature adsorber. The low-pressure stage begins on switching to the low-temperature heat reservoir. At this stage, the process of adsorption occurs in the high-temperature reactor, and the process of desorption occurs in the low-temperature adsorber. This process is endothermic, which contributes to the cooling of the low-temperature adsorber. Now we consider the case with condensation. For this case, taking account of phase transitions in the low-temperature adsorber, we increase the capacity of the high-temperature reactor fourfold. Figure 8 plots the temperatures in the adsorbers and the temperatures of the heat-transfer agents at the reactor outlet versus time with account of condensation in meso- and macropores. In this case, as in the previous one, the operation of the refrigerator includes the steps of high and low pressures. The only difference is that new processes, i.e., condensation and evaporation in the low-temperature adsorber, are added to the operating cycle of the converter. Thus, each stage of the operating cycle may be subdivided into two steps. The 1268
Fig. 8. Time dependences of the temperatures in the adsorbers averaged over the reactor volume (a) and of the temperatures of the heat-transfer agents at the reactor outlet (b) for the case with condensation.
TABLE 1. Basic Characteristics of Performance of the Adsorption Thermal-Energy Converter for the Regimes with Condensation and without Condensation Lhigh/Llow
COP
SCP, W/kg
ΔT , оС
Without condensation
1
0.54
81.1
8.0
With condensation
4
0.28
98.4
15.8
Regime
first step of the high-pressure stage corresponds to region 1 in Fig. 8. In this step, we have the process of adsorption in the low-temperature reactor, after which condensation begins in mesopores and thereafter in macropores, which corresponds to the second step of the stage in question (region 2). We have an analogous situation for the next stage, i.e., the low-pressure stage which includes evaporation (region 3) and desorption (region 4) processes. Switchings between the stages are effected when the same condition as in the previous case is fulfilled. However, it should be noted that no strict subdivision exists between steps 1 and 2 and between 3 and 4, since, e.g., in operation of the refrigerator at the high-pressure stage, a situation appears where condensation has already begun in the central part of the reactor, whereas the process of adsorption is still in progress at its periphery. It is clear from the two considered cases that the process of condensation and evaporation increases the cycle time, which may already adversely affect the SCP. Also, we note that in the case of condensation the maximum decrease in the temperature of the heat-transfer agent at the reactor outlet in the low-pressure stage is 13оС larger than in the case without condensation. Table 1 gives the basic characteristics of performance of the adsorption refrigerator for the considered cases. The data in this table point to the circumstance that with introduction of the process of condensation into the operation of the adsorption refrigerator, the difference of the temperatures of the low-temperature heat-transfer agent at the inlet and outlet of the reactor, which is averaged over the time of the low-pressure stage, increases. Also, the SCP growth is tracked. As far as the COP is concerned, it substantially drops for the case with condensation. As has already been mentioned, the process of condensation can be controlled variously: by changing different parameters of the system or by creating certain conditions of switching of the stages in the cycle. Of interest is the manner in which any of these methods influences the performance of the adsorption refrigerator. For this purpose we consider a few cases illustrating the influence of the change in the system’s parameters on the basic characteristics of heat-conversion efficiency. Figure 9 gives the dependences of the effective parameters of operation of the adsorption refrigerator on the relation of the capacities of the high-temperature and low-temperature adsorbers for two different temperatures of the high-temperature heat reservoir: 170 and 140 оС. Utilization of the high-temperature heat-transfer agent is seen to allow the complete 1269
Fig. 9. Characteristics of performance of the adsorption refrigerator vs. relation of the capacities of the high-temperature and low-temperature adsorbers at Thigh = 170 (a) and 140оС (b).
TABLE 2. Change in the Integral Heat of Desorption, Heat of Evaporation, and the Total Heat of Cooling as a Function of the Relation of the Adsorber Capacities Lhigh/Llow
1
1.5
2
2.5
3
3.5
4
Qdes, kJ
262
260
224
182
140
102
73
Qev, kJ
0
78
184
292
400
505
572
Qtot, kJ
262
338
408
474
540
607
645
condensation in all pores of the sorbent of the low-temperature reactor. As far as the case of the high-temperature reservoir with a lower temperature is concerned, filling of the sorbent pores with condensate is incomplete in refrigerator operation: mesopores are filled totally, whereas macropores are filled partially. Also, it should be noted that in this case, for a certain relation of the capacities of the high-temperature and low-temperature adsorbers, the performance characteristics change the direction of their variation (in terms of growth or decrease). This is a consequence of the "poor" matching of the capacities of the high-temperature and low-temperature reactors, since a much greater amount of the adsorbate than needed for participa-
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tion in each stage of the operating cycle of the adsorption refrigerator is brought as a result into the high-temperature adsorber. The given plots illustrate the reduction in the COP and the growth in the average cooling temperature with increase in the content of the condensate in the pores of the low-temperature sorbent. Increase in the temperature of the high-temperature heat-transfer agent increases the share of the condensate in the sorbent pores, which contributes to the growth in the performance characteristics, except for the COP. With increase in the share of the condensate in the sorbent pores, the SCP behaves differently at different temperatures of the high-temperature heat-transfer agent. Thus, at higher Thigh, this characteristic grows with the content of the condensed sorbate in the low-temperature adsorber and decreases at lower Thigh. This is due to the fact that at large values of the temperature of the high-temperature reservoir, the processes in the reactors are faster; therefore, the duration of the cycle is reduced and hence phase transition of the sorbate leads to a considerable (more than twofold) increase in the total quantity of heat absorbed in the low-temperature unit. Actually, in the considered limiting case where the pore volume (meso- and macropores) of the low-temperature unit is completely filled with condensate, this unit operates mainly as a condenser and an evaporator, whereas the high-temperature unit preserves its functions as an adsorber. Conclusions. The performed modeling of the operation of an adsorption thermal-energy converter with allowance for the possibility of phase transitions in sorbent pores has enabled us to investigate the influence of the processes of condensation and evaporation on the performance of such devices. The operating characteristics of the system may vary with the method of control of condensation. In all the considered cases we observed a growth in the difference of the temperatures of the heattransfer agent at the reactor inlet and outlet, averaged over the cooling-stage time, and a decrease in the COP with increase in the share of the condensate in the sorbent pores. The SCP increased with the share of the condensate in the low-temperature adsorber in all cases, except for the case where the temperature of the high-temperature reservoir was low. It has been shown that the fraction of the sorbent pores filled with the adsorbate that condensed in the operating cycle of the adsorption refrigerator depends on the selection of the temperature of the high-temperature reservoir. Thus, at higher temperatures of the high-temperature heat-transfer agent, the low-temperature adsorber was completely filled with condensate, whereas at low temperatures it was partially filled. Important factors influencing the performance of adsorption heat converters are the conditions of switching of cyclic stages and matching of the capacities of the low-temperature and high-temperature adsorbers with allowance for the processes to be implemented inside each unit. It is significant that when condensation is realized in the low-temperature adsorber the capacity of the high-temperature reactor does not exceed a certain optimum value — the total capacity of the low-temperature unit — since otherwise, all characteristics of the system decline. Also, it should be noted that the change in the relation between the volumes of meso- and macropores, with their total volume remaining constant (for the selected parameters of the system and conditions of switching of cyclic stages), does not influence, in practice, the effective parameters of the adsorption thermal-energy converter.
NOTATION A and C, constants of the kinetics of chemical adsorption, s–1; a and amax, adsorption and limiting value of adsorption, kg/kg; B, constant dependent on the micropore size; cP, specific heat at constant pressure, J/ (kg⋅K); d and deff, diameter and effective diameter of the pores, m; E, activation energy, J/kg; Gf, mass flow rate of the heat-transfer agent, kg/s; ΔH, enthalpy, J/mole; Jgas-liq and Qgas-liq, rate and heat of phase transition, kg/(s⋅m3) and J; L, adsorber length, m; ms, mass of the sorbent, kg; M, molar mass, kg/mole; P, pressure, Pa; P*, P′, Ph and Pσ, pressure in the neighboring reactor, steam pressure above the concave surface, hydrostatic pressure, and capillary pressure, Pa; Qs, sorption heat, J; R and R0, external and internal radii of the adsorber, m; Rg, universal gas constant, J/(mole⋅K); r, radial constant, m; rp, pore radius, m; S, heat-transfer surface, m2; Sgas-liq, specific condensation and evaporation surface, m–1; ΔS, entropy, J/(mole⋅K); T, temperature, K; T and ΔT , volume-averaged temperature in the adsorber and average cooling temperature, K; t, time, s; V, volume, m3; Vliq, molar volume of liquid ammonia, m3/mole; v, gas velocity, m/s; W, adsorption rate, s–1; W0, limiting adsorption-space volume expressing the micropore volume; x, degree of completeness of the reaction; α, heat-transfer coefficient, W/(K⋅m2); β, affinity coefficient; ε, porosity; θ, saturation; λ, thermal conductivity, W/(K⋅m); ρ, density, kg/ m3; σ, surface-tension coefficient. Subscripts: a, adsorbate; ads, adsorption; ch, chemical; cr, critical; dec, decomposition; des, desorption; eff, effective; env, environment; eq, equilibrium; f, fluid (heat-transfer agent); f,in and f,out, heat-transfer agent at the reactor inlet and outlet; g, gas; high and low, parameters referring to the high-temperature and low-temperature heat-transfer agent or adsorbers; liq, liquid; mac, macropore; mes, mesopore; p, pore; s, sorbent; sat, saturation; syn, synthesis; thr, throttle; tot, total; w, interior adsorber wall; ch, chemical adsorption; ph, physical adsorption; cyc, cycle.
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